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in Problem solving
I am certain from your description that what you are describing is indeed ivy. what may be throwing you is that ivy doesn't flower and fruit while it is in upward climbing mode, but once it gets to the top of whatever it is clinging to then it flops and produces a different shaped leaf altogether, quite unlike the earlier form. And it produces its fruit in winter, great for these fruits are exactly how you describe them, clusters of green balls with black eyes.
Here's a close up of ivy fruit:
and here's a photo of an adult ivy leaf with tiny unripened fruit.
You can trim it to make it tidy looking but removing Ivy can be a pain. it is actually very good for wildlife.
I'd remove it. Cut to the base, treat any new soft leaf growth with glyphosate. You will need to do this on several occasions to exhaust the roots. I know some will say you should leave it as it provides food for birds at a time when food is scarce, but it can be very unruly in a garden and can spoil your garden plants / shrubs. Maybe you have an uninteresting tree somewhere where you could leave some ivy?